puolisuunnikasmenetelmässä

Puolisuunnikasmenetelmä, also known as the trapezoidal rule, is a numerical integration technique used to approximate the definite integral of a function. It is based on approximating the area under the curve of the function by dividing it into a series of trapezoids. The area of each trapezoid is calculated and then summed up to obtain an approximation of the total integral.

To apply the puolisuunnikasmenetelmä, the interval of integration is divided into n subintervals of equal width, denoted by h. The width of each subinterval is calculated as h = (b - a) / n, where a and b are the lower and upper limits of integration, respectively. The function values at the endpoints of each subinterval are then used to form the parallel sides of a trapezoid, and the width of the subinterval serves as the height of the trapezoid.

The area of a single trapezoid is given by the formula A = (f(x_i) + f(x_{i+1})) * h / 2, where x_i and x_{i+1} are the endpoints of the i-th subinterval. The total approximate integral is the sum of the areas of all n trapezoids: Integral approx = h/2 * [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]. As the number of subintervals n increases, the approximation generally becomes more accurate.

The puolisuunnikasmenetelmä is a relatively simple and widely used numerical method for approximating integrals, particularly when an analytical solution is difficult or impossible to obtain. It is a foundational concept in numerical analysis and forms the basis for more advanced integration techniques.